Questions tagged [line-bundles]

A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.

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6 votes
1 answer
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Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor

Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...
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1 vote
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Bundles vs. line bundles

Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...
2 votes
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145 views

Semi-continuity of the Picard number

Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$. ...
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1 vote
1 answer
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Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve

We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
1 vote
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Representatives of line bundle cohomology over tori

Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
5 votes
1 answer
190 views

Extension of first order deformations of a line bundle

Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
4 votes
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168 views

Ample divisor of degree two on a blow-up of $\mathbb P^2$ at nine points

Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position. I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$). Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \...
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1 vote
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Extension of a section of a line bundle on a family of curves to the central fibre

I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line ...
2 votes
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76 views

Linear system of a relative effective divisor on an arithmetic surface contains vertical divisors

I am puzzled by the behavior of some divisors in my attempt to understand the relative Picard functor $\mathrm{Pic}_{X/S}$ of an arithmetic surface $\pi:X\to S$. This is defined by relative divisors $...
2 votes
1 answer
142 views

Morphism attached to a big and globally generated line bundle

Let $X$ be a smooth projective variety over a field $k$. Let $L$ be an invertible sheaf on $X$. Suppose that $L$ is big and globally generated. Can one conclude that the associated morphism $\phi_L : ...
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1 answer
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Compact complex non-Kähler manifolds with nef canonical bundle

Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples? Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...
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1 answer
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Arithmetic ampleness and scalings of the metric

Let $\overline L= (L, h)$ be a hermitian $C^ \infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & ...
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4 votes
1 answer
174 views

Existence of non-trivial "line-symplectic" manifolds

One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...
2 votes
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Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification

Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better. In Liu-Lorenzini-Raynaud (...
4 votes
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353 views

Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
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3 votes
1 answer
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When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?

This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question. Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
2 votes
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Conditions for long exact sequence for line bundles on curve to degenerate?

Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$. The sequence $$0\to \mathcal{L}' \xrightarrow{...
4 votes
1 answer
215 views

Type vs degree of a polarized abelian variety

Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that $d = \chi(L) = \dim H^0(A,L)$ since $L$ is ample. I've read in a lot ...
3 votes
1 answer
209 views

Global choice of eigenvectors on an open surface

Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...
5 votes
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196 views

Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold. The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
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2 votes
1 answer
82 views

Grassmannian of line subbundle of a stable rank 2 vector bundle on a smooth projective curve

Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $...
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1 vote
1 answer
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Riemannian vector bundle [closed]

I'm trying to show the curvature of a one dimensional vector bundle with a Riemannian metric vanishes, no matter what the connection is. I found this can be done for orientable bundles, because an ...
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3 votes
1 answer
223 views

Galois invariant line bundle and base change

Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
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Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
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3 votes
2 answers
227 views

Character which defines canonical bundle on flag variety

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
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1 vote
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102 views

From a factor of automorphy on an abelian variety to a divisor

Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
1 vote
1 answer
281 views

Making a vector bundle ample by twisting with ample line bundle

Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is ample if the relative twisting sheaf $\mathcal{O}_{\mathbf{P}(E)...
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1 vote
0 answers
229 views

Construction of the Hilbert Scheme

I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this: Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
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1 answer
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Holomorphic sections to anti-holomorphic sections

Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am trying to give a norm preserving isomorphism between the space of holomorphic ...
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Classification of square roots of line bundles and metalinear/metaplectic structures

Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
1 vote
1 answer
195 views

Flat connection of a degree zero line bundle on curve

The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...
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1 vote
1 answer
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(Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?

I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper, https://arxiv.org/abs/math/0411613. However I cannot purpose analogously on $D^b(P^1 \times P^2)$. ...
4 votes
0 answers
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Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers. Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...
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Proof of uniqueness in the universal property of Poincaré line bundles

My question concerns the proof of a part of Lemma IV.2.2 (pag. 168) of the book Geometry of Algebraic Curves. vol. I by Arbarello, Cornalba, Griffiths and Harris. In order to state my problem, let me ...
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4 votes
1 answer
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Identity relating iterated determinant line bundles

Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
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Sections of vector bundles interpreted as sections of line bundles

Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...
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1 vote
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Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
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Discriminant divisor $\mathcal{D}_{r} \subseteq H^{0}(X,K_{X}^{\otimes r})$ is irreducible

Let $X\colon$ smooth projective curve over $\mathbb{C}$, $K_{X}\colon$ canonical line bundle over $X$, and $W_{r}$ denotes $H^{0}(X,K_{X}^{\otimes r})$. I'm trying to prove the following proposition, ...
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What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?

Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...
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9 votes
2 answers
311 views

Differential refinement of homology

Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
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1 vote
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Vector bundle defined by using divisors of very ample line bundle

Let $X$ be a smooth projective curve. Suppose that $L_1$ and $L_2$ are line bundles on $X$, and $L_1$ is very ample. $\operatorname{Div}(s)$ denotes a divisor defined by a global section $s\in H^0(X,L)...
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7 votes
1 answer
242 views

The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
1 vote
0 answers
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Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
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7 votes
1 answer
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Multiplication maps for big line bundles

In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line ...
5 votes
1 answer
410 views

Line bundle on product scheme

Let $k$ be a field, $X$ be a complete variety over $k$, $V$ be an open subvariety of $X$, $Y$ be a scheme over $k$. Suppose $L$ is a line bundle on $V\times Y$. If $L|_{V\times\lbrace y\rbrace}$ ...
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3 votes
1 answer
118 views

Weights on the linearization

Consider, just as an example, an action of $\mathbb{C}^*$ on $\mathbb{P}^2$ of the form $$t\cdot p=[p_0:tp_1:t^2p_2]$$ There are $3$ fixed points, namely $e_1,e_2,e_3$. If I consider a $\mathbb{C}^*$-...
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3 votes
1 answer
392 views

Weak Lefschetz theorem for Lef line bundles

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following....
1 vote
0 answers
126 views

Problem regarding existence of a divisor representing line bundle

We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...
1 vote
0 answers
162 views

The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...
2 votes
1 answer
232 views

Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...

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